PKSFC package

• Allowing to simulate SFC models in an open source environment}
• Still preliminary
• Only one numerical solver: Gauss-Seidel algorithm (Kinsella and O’Shea 2010)
• github.com/s120/pksfc
• Technical aspect
• R package
• EViews translator
• Read equation files
• Build model from console
• Visualization/Design tools and helpers

Installing the dependent libraries and the package

You need to install all the required libraries This is for traditional libraries

install.packages("expm")
install.packages("igraph")

For non-conventional libraries, such as the one need to visualize Direct Acyclical Graphs (DAG), you need to do the following

source("http://bioconductor.org/biocLite.R")
biocLite("Rgraphviz")

Finally you can then download the PKSFC package from github and install it locally

install.packages("path/PKSFC_1.5.tar.gz", repos = NULL, 
                                 type="source")

Testing

Now we’re ready to load the package:

library(PKSFC)

## Loading required package: expm

## Loading required package: Matrix

## 
## Attaching package: 'expm'

## The following object is masked from 'package:Matrix':
## 
##     expm

## Loading required package: igraph

## 
## Attaching package: 'igraph'

## The following objects are masked from 'package:stats':
## 
##     decompose, spectrum

## The following object is masked from 'package:base':
## 
##     union

1. Load SIM (download SIM.txt from Github)

sim<-sfc.model("Models/SIM.txt",modelName="SIMplest model 
        from chapter 3 of Godley and Lavoie (2007)")

2. Simulate the model

datasim<-simulate(sim)

3. replicate figure 3.2 of page 73.

matplot(sim$time,datasim$baseline[,c("Yd","C")],type="l", 
    xlab="", ylab="",lwd=2,lty=c(2,3))
lines(sim$time,vector(length=length(sim$time))
    +datasim$baseline["2010","C"], lwd=2)
legend(x=1970,y=50,lwd=2, legend=c("Disposable Income",
    "Consumption","Steady State"),lty=c(2,3,1), bty="n")

How does it work?

• The package parses a text file containing the equations
• It generates an internal representation of the model
• It checks the internal consistency of the model, the calibration
• Allows to simulate the model using a linear solver, the Gauss-Seidel Algorithm

Source code of SIM

#1. EQUATIONS
C = C
G = G
T = T
N = N
Yd = W*N - T
T = theta*W*N
C = alpha1*Yd + alpha2*H_h(-1)
H = H(-1) + G - T
H_h = H_h(-1) + Yd - C
Y = C + G
N = Y/W
#2. PARAMETERS
alpha1=0.6
alpha2=0.4
theta=0.2
#EXOGENOUS
G=20
W=1
#INITIAL VALUES
H=0
H_h=0
#3. Timeline
timeline 1945 2010

A few important points regarding the model source code:

• The first line of the code should be a comment line (starting with #)
• The file should not contain any empty lines.
• You should avoid naming your variables with reserved names in R such as ‘in’ or ‘max’.
• There should be only one equation per line.
• There should be only one variable on the left hand side of the equation.
• You can use R functions such as min, max, or logical operators such as > or <=. In the case the logical operator returns true, the numeric value will be one. Thus (100>10) will return 1.
• The lag operator is represented by (-x) where x is the lag.
• You can add as many comments, using the # character at the beginning of the line. Each comment exactly above an equation will be considered as the description of the equation and will be stored in the internal representation of the sfc model object.

Internal representation

print(sim)

## $name
## [1] "SIMplest model \n\t\tfrom chapter 3 of Godley and Lavoie (2007)"
## 
## $simulated
## [1] FALSE
## 
## $variables
##       name     initial value description    
##  [1,] "Yd"     NA            "1. EQUATIONS" 
##  [2,] "T"      NA            ""             
##  [3,] "C"      NA            ""             
##  [4,] "H"      "0"           ""             
##  [5,] "H_h"    "0"           ""             
##  [6,] "Y"      NA            ""             
##  [7,] "N"      NA            ""             
##  [8,] "alpha1" "0.6"         "2. PARAMETERS"
##  [9,] "alpha2" "0.4"         ""             
## [10,] "theta"  "0.2"         ""             
## [11,] "G"      "20"          "EXOGENOUS"    
## [12,] "W"      "1"           ""             
## 
## $endogenous
##      name  initial value lag description   
## [1,] "Yd"  NA            "0" "1. EQUATIONS"
## [2,] "T"   NA            "0" ""            
## [3,] "C"   NA            "0" ""            
## [4,] "H"   "0"           "1" ""            
## [5,] "H_h" "0"           "1" ""            
## [6,] "Y"   NA            "0" ""            
## [7,] "N"   NA            "0" ""            
## 
## $equations
##      endogenous value equation                 description   
## [1,] "Yd"             "W*N-T"                  "1. EQUATIONS"
## [2,] "T"              "theta*W*N"              ""            
## [3,] "C"              "alpha1*Yd+alpha2*H_h_1" ""            
## [4,] "H"              "H_1+G-T"                ""            
## [5,] "H_h"            "H_h_1+Yd-C"             ""            
## [6,] "Y"              "C+G"                    ""            
## [7,] "N"              "Y/W"                    ""            
## 
## $time
##  [1] 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958
## [15] 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972
## [29] 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986
## [43] 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
## [57] 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
## 
## $matrix
##     Yd T C H H_h Y N
## Yd   0 1 0 0   0 0 1
## T    0 0 0 0   0 0 1
## C    1 0 0 0   0 0 0
## H    0 1 0 0   0 0 0
## H_h  1 0 1 0   0 0 0
## Y    0 0 1 0   0 0 0
## N    0 0 0 0   0 1 0
## 
## $blocks
## $blocks[[1]]
## [1] 2 3 4 6 7 1 5
## 
## 
## attr(,"class")
## [1] "sfc"

Output data structure

• Output is a list of matrix where each element of the list are a scenario
  • baseline

  • scenario_i

    Y	d	T	C	H H_	h	Y	N alpha1	alpha2	theta	G W	it	er bl	ock 1
    1945	NA	NA	NA	0.00000	0.00000	NA	NA	0.6	0.4	0.2	20	1	0
    1946	30.76923	7.692308	18.46154	12.30769	12.30769	38.46154	38.46154	0.6	0.4	0.2	20	1	293
    1947	38.34320	9.585799	27.92899	22.72189	22.72189	47.92899	47.92899	0.6	0.4	0.2	20	1	269
    1948	44.75193	11.187984	35.93992	31.53391	31.53391	55.93992	55.93992	0.6	0.4	0.2	20	1	263
    1949	50.17471	12.543678	42.71839	38.99023	38.99023	62.71839	62.71839	0.6	0.4	0.2	20	1	258
    1950	54.76322	13.690805	48.45402	45.29943	45.29943	68.45402	68.45402	0.6	0.4	0.2	20	1	254
    1951	58.64580	14.661450	53.30725	50.63798	50.63798	73.30725	73.30725	0.6	0.4	0.2	20	1	251
    1952	61.93106	15.482766	57.41383	55.15521	55.15521	77.41383	77.41383	0.6	0.4	0.2	20	1	247
    1953	64.71090	16.177725	60.88862	58.97749	58.97749	80.88862	80.88862	0.6	0.4	0.2	20	1	244
    1954	67.06307	16.765767	63.82884	62.21172	62.21172	83.82884	83.82884	0.6	0.4	0.2	20	1	242
    1955	69.05337	17.263341	66.31671	64.94838	64.94838	86.31671	86.31671	0.6	0.4	0.2	20	1	239
    1956	70.73746	17.684366	68.42183	67.26401	67.26401	88.42183	88.42183	0.6	0.4	0.2	20	1	236
    1957	72.16247	18.040617	70.20309	69.22339	69.22339	90.20309	90.20309	0.6	0.4	0.2	20	1	234
    1958	73.36824	18.342061	71.71030	70.88133	70.88133	91.71030	91.71030	0.6	0.4	0.2	20	1	231
    1959	74.38851	18.597128	72.98564	72.28421	72.28421	92.98564	92.98564	0.6	0.4	0.2	20	1	229
    1960	75.25182	18.812955	74.06477	73.47125	73.47125	94.06477	94.06477	0.6	0.4	0.2	20	1	227
    1961	75.98231	18.995577	74.97789	74.47567	74.47567	94.97789	94.97789	0.6	0.4	0.2	20	1	224
    1962	76.60041	19.150104	75.75052	75.32557	75.32557	95.75052	95.75052	0.6	0.4	0.2	20	1	222
    1963	77.12343	19.280857	76.40428	76.04471	76.04471	96.40428	96.40428	0.6	0.4	0.2	20	1	220
    1964	77.56598	19.391494	76.95747	76.65322	76.65322	96.95747	96.95747	0.6	0.4	0.2	20	1	218
    1965	77.94044	19.485111	77.42555	77.16811	77.16811	97.42555	97.42555	0.6	0.4	0.2	20	1	215
    1966	78.25730	19.564324	77.82162	77.60378	77.60378	97.82162	97.82162	0.6	0.4	0.2	20	1	213
    1967	78.52541	19.631351	78.15676	77.97243	77.97243	98.15676	98.15676	0.6	0.4	0.2	20	1	211
    1968	78.75227	19.688067	78.44033	78.28437	78.28437	98.44033	98.44033	0.6	0.4	0.2	20	1	209
    1969	78.94423	19.736056	78.68028	78.54831	78.54831	98.68028	98.68028	0.6	0.4	0.2	20	1	207
    1970	79.10665	19.776663	78.88332	78.77165	78.77165	98.88332	98.88332	0.6	0.4	0.2	20	1	205
    1971	79.24409	19.811023	79.05511	78.96062	78.96062	99.05511	99.05511	0.6	0.4	0.2	20	1	202
    1972	79.36038	19.840096	79.20048	79.12053	79.12053	99.20048	99.20048	0.6	0.4	0.2	20	1	200
    1973	79.45879	19.864697	79.32348	79.25583	79.25583	99.32348	99.32348	0.6	0.4	0.2	20	1	198
    1974	79.54205	19.885513	79.42756	79.37032	79.37032	99.42756	99.42756	0.6	0.4	0.2	20	1	196
    1975	79.61250	19.903126	79.51563	79.46719	79.46719	99.51563	99.51563	0.6	0.4	0.2	20	1	194
    1976	79.67212	19.918030	79.59015	79.54916	79.54916	99.59015	99.59015	0.6	0.4	0.2	20	1	192
    1977	79.72256	19.930640	79.65320	79.61852	79.61852	99.65320	99.65320	0.6	0.4	0.2	20	1	190
    1978	79.76524	19.941311	79.70656	79.67721	79.67721	99.70656	99.70656	0.6	0.4	0.2	20	1	187
    1979	79.80136	19.950340	79.75170	79.72687	79.72687	99.75170	99.75170	0.6	0.4	0.2	20	1	185
    1980	79.83192	19.957980	79.78990	79.76889	79.76889	99.78990	99.78990	0.6	0.4	0.2	20	1	183
    1981	79.85778	19.964445	79.82222	79.80445	79.80445	99.82222	99.82222	0.6	0.4	0.2	20	1	181
    1982	79.87966	19.969915	79.84957	79.83453	79.83453	99.84957	99.84957	0.6	0.4	0.2	20	1	179
    1983	79.89817	19.974543	79.87272	79.85999	79.85999	99.87272	99.87272	0.6	0.4	0.2	20	1	177
    1984	79.91384	19.978460	79.89230	79.88153	79.88153	99.89230	99.89230	0.6	0.4	0.2	20	1	175
    1985	79.92709	19.981774	79.90887	79.89975	79.89975	99.90887	99.90887	0.6	0.4	0.2	20	1	173
    1986	79.93831	19.984578	79.92289	79.91518	79.91518	99.92289	99.92289	0.6	0.4	0.2	20	1	170
    1987	79.94780	19.986950	79.93475	79.92823	79.92823	99.93475	99.93475	0.6	0.4	0.2	20	1	168
    1988	79.95583	19.988958	79.94479	79.93927	79.93927	99.94479	99.94479	0.6	0.4	0.2	20	1	166
    1989	79.96263	19.990657	79.95328	79.94861	79.94861	99.95328	99.95328	0.6	0.4	0.2	20	1	164
    1990	79.96838	19.992094	79.96047	79.95652	79.95652	99.96047	99.96047	0.6	0.4	0.2	20	1	162
    1991	79.97324	19.993310	79.96655	79.96321	79.96321	99.96655	99.96655	0.6	0.4	0.2	20	1	160
    1992	79.97736	19.994340	79.97170	79.96887	79.96887	99.97170	99.97170	0.6	0.4	0.2	20	1	158
    1993	79.98084	19.995210	79.97605	79.97366	79.97366	99.97605	99.97605	0.6	0.4	0.2	20	1	156
    1994	79.98379	19.995947	79.97974	79.97771	79.97771	99.97974	99.97974	0.6	0.4	0.2	20	1	153
    1995	79.98628	19.996571	79.98285	79.98114	79.98114	99.98285	99.98285	0.6	0.4	0.2	20	1	151
    1996	79.98839	19.997098	79.98549	79.98404	79.98404	99.98549	99.98549	0.6	0.4	0.2	20	1	149
    1997	79.99018	19.997545	79.98772	79.98650	79.98650	99.98772	99.98772	0.6	0.4	0.2	20	1	147
    1998	79.99169	19.997923	79.98961	79.98857	79.98857	99.98961	99.98961	0.6	0.4	0.2	20	1	145
    1999	79.99297	19.998242	79.99121	79.99033	79.99033	99.99121	99.99121	0.6	0.4	0.2	20	1	143
    2000	79.99405	19.998513	79.99256	79.99182	79.99182	99.99256	99.99256	0.6	0.4	0.2	20	1	141
    2001	79.99497	19.998741	79.99371	79.99308	79.99308	99.99371	99.99371	0.6	0.4	0.2	20	1	139
    2002	79.99574	19.998935	79.99468	79.99414	79.99414	99.99468	99.99468	0.6	0.4	0.2	20	1	136
    2003	79.99640	19.999099	79.99549	79.99504	79.99504	99.99549	99.99549	0.6	0.4	0.2	20	1	134
    2004	79.99695	19.999237	79.99619	79.99581	79.99581	99.99619	99.99619	0.6	0.4	0.2	20	1	132
    2005	79.99742	19.999355	79.99677	79.99645	79.99645	99.99677	99.99677	0.6	0.4	0.2	20	1	130
    2006	79.99782	19.999454	79.99727	79.99700	79.99700	99.99727	99.99727	0.6	0.4	0.2	20	1	128
    2007	79.99815	19.999538	79.99769	79.99746	79.99746	99.99769	99.99769	0.6	0.4	0.2	20	1	126
    2008	79.99844	19.999609	79.99805	79.99785	79.99785	99.99805	99.99805	0.6	0.4	0.2	20	1	124
    2009	79.99868	19.999669	79.99835	79.99818	79.99818	99.99835	99.99835	0.6	0.4	0.2	20	1	122
    2010	79.99888	19.999720	79.99860	79.99846	79.99846	99.99860	99.99860	0.6	0.4	0.2	20	1	119

The Gauss Seidel Algorithm

• Principle: Solving \(Ax=b,\, A\in\mathbb{R}^{n\times n},\, b\in \mathbb{R}^n\) via an iterative algorithm, where each iteration can be represented by \(L x^{k+1} = b-Ux^{k},\, A=L+U\). Where \(L\) is lower triangular and \(U\) is upper triangular.

• Pseudo-code:

1. Select initial values \(x^0\)
2. While \(k<maxIter\) & \(\delta < tolValue\)
   1. For each \(i=1,...,n\): \[x_i^{k+1}=\frac{1}{a_{ii}}\left( b_i-\sum^{i-1}_{j=1}a_{ij} x_j^{k+1}-\sum_{j=i+1}^n a_{ij}x_j^k\right)\]
   2. Compute \(\delta\): \[\delta = \frac{x^{k+1}-x^k}{x^k}\]

System of (in)dependent equations

See Fenell et. al (2016)

• The Gauss-Seidel has to be used only in the case of system of dependent equations
• In other case, we only need to find order in which each variable is computed and simply compute the new value in each period
• This order is the “logical causal order” of the model and can be visualized using Direct (A)Cyclical Graphs

Direct Acyclic graphs

simex<-sfc.model("Models/SIMEX.txt",modelName="SIMplest model with expectation")
layout(matrix(c(1,2),1,2))
plot.dag(sim,main="SIM" )
plot.dag(simex,main="SIMEX" )

• Aside from the mathematical implication that a system of equation represent, it also has an economic meaning:
• the economy represented by SIM will adjust in one period to any shock applied to government spending.
• for SIMEX it is not the case because consumption depend on expected disposable income which is equal to previous period disposable income.
• in this case, the economy represented by SIMEX will adjust slowly to a shock applied to government spending, via the stocks (and particularly the buffer stock)

In the case of a more complex model - Chapter 6

• We can generate the plots that allow us to delve into the actual structure of the system.
• Nodes that do not form a cycle are green while nodes that form a cycle in the system are pink.
• The Gauss-Seidel needs to be applied only for the cycles

ch6 <- sfc.model("Models/ch6.txt",modelName="Chapter6_openmodel")
graphs = generate.DAG.collapse( adjacency = ch6$matrix )
par(mfrow = c(1,3))
# first plot the orgianl grpah
plot_graph_hierarchy( graphs$orginal_graph, main = "orginal graph" )
# plot hte nodes that form the strongly connected compoent
plot_graph_hierarchy( graphs$SCC_graph, main = "SCC nodes" )
# plot the result ing DAG when we take the condensation of the graph
plot_graph_hierarchy( graphs$DAG, main = "resulting DAG" )

Systems of dependent vs independent equations

#Doing the simulations
datasimex<-simulate(simex)
init = datasimex$baseline[66,]
simex<-sfc.addScenario(simex,"G",25,1960,2010,init)
datasimex<-simulate(simex)
datasim<-simulate(sim)
init = datasim$baseline[66,]
sim<-sfc.addScenario(sim,"G",25,1960,2010,init)
datasim<-simulate(sim)

#Plotting it all
layout(matrix(c(1,2),1,2))
matplot(sim$time,datasim$scenario_1[,c("H","C","Yd")],
    lty=1:3,type="l",xlab="",ylab="",main="SIM")
legend(x=1944,y=130,legend=c("Wealth","Consumption",
    "Disposable Income"), lty=1:3,bty="n")
matplot(simex$time,datasimex$scenario_1[,c("H","C","Yd",
    "Yd_e")],lty=1:4,type="l",xlab="",ylab="",main="SIMEX")
legend(x=1944,y=130,legend=c("Wealth","Consumption","Disposable Income",
   "Expecetd Disposable Income"),lty=1:4,bty="n")

Computational implications

Let’s see how much time it takes to run sim:

ptm <- proc.time()
data1<-simulate(sim)
print(paste("Elapsed time is ",proc.time()[3]-ptm[3],
                        "seconds"))

## [1] "Elapsed time is  3.67 seconds"

Now lets play with some of the parameters of the simulate function:

1. tolValue

ptm <- proc.time()
data2<-simulate(sim,tolValue = 1e-3)
print(paste("Elapsed time is ",proc.time()[3]-ptm[3],"seconds"))

## [1] "Elapsed time is  0.176000000000002 seconds"

2. maxIter

ptm <- proc.time()
data3<-simulate(sim, maxIter=10)
print(paste("Elapsed time is ",proc.time()[3]-ptm[3],"seconds"))

## [1] "Elapsed time is  0.173999999999999 seconds"

Observing the results of the three simulations

kable(round(t(cbind(data1$baseline[c(1,2,20,40,66),c("Y")],
        data2$baseline[c(1,2,20,40,66),c("Y")],
        data3$baseline[c(1,2,20,40,66),c("Y")])),digits=3))

Block Gauss-Seidel

The order of equations matters, if first compute variables that do not depend on current period, this speeds the process. Define blocks of equation independent from the others.

print(simex$blocks)

## [[1]]
## [1] 7
## 
## [[2]]
## [1] 3
## 
## [[3]]
## [1] 6
## 
## [[4]]
## [1] 8
## 
## [[5]]
## [1] 2
## 
## [[6]]
## [1] 1 4
## 
## [[7]]
## [1] 5

Simulation of SIMEX

ptm <- proc.time()
dataex<-simulate(simex,tolValue = 1e-10)
print(paste("Elapsed time is ",proc.time()[3]-ptm[3]
                        ,"seconds"))

## [1] "Elapsed time is  0.117999999999999 seconds"

Results for SIMEX

kable(round(t(dataex$baseline[c(1,2,20,40,66),c("G","Y",
    "T","Yd","Yd_e","C","H","H_h")]),digits=3))

Output data structure

• Output is a list of matrix where each element of the list are a scenario
• baseline
• scenario_i
• In the result matrix, there is a column indicating the number of iteration in the Gauss-Seidel algorithm per block of equations per period

kable(dataex$baseline)

Checking the number of iteractions for SIM

kable(round(t(cbind(
    data1$baseline[c(1,2,20,40,66),c("iter block 1")],
    data2$baseline[c(1,2,20,40,66),c("iter block 1")],
    data3$baseline[c(1,2,20,40,66),c("iter block 1")]
    )),digits=3))

Checking the number of iteractions for simex, no simulate parameters

kable(round(t(dataex$baseline[c(1,2,20,40,66),
    c("iter block 1","iter block 2","iter block 3",
    "iter block 4","iter block 5","iter block 6",
    "iter block 7")]),digits=3))

Visualisation

• Interactions with economists (mainstream and less)
  • Importance of stock and flow dimensions (Bezemer et al. 2015)
  • Importance of clearing mechanisms for markets under uncertainty (Foley 1975)
  • Importance of disequilibrium mechanics, path dependency, etc…
• Interactions with students
  • Undergrad level, with dynamic models without coding (Shiny, Mathematica, …): simple pK results such as endogenous money, paradox of thrift, Minskian dynamics, etc.
  • Postgraduate level, integrated with R: sectoral accounts (empirical), mathematical properties (causal structure, solutions), more complex pK results and model building (PKSFC package)
• Interactions with policy makers
  • Importance of visualization (results, causal structure)
  • Importance of interactions with the model
  • Importance of real-world structures

Visualisation - Sankey Diagram

Flows in the UK Economy 2014, source: Burgess et al. (2016)

Visualisation - Graph representation

• Fenell et al. (2015) show that each SFC model can be perceived as a Direct Acyclic Graph, which allows to understand the internal causal structure of the model (imposed).
• Practically, all endogenous variables are nodes and there if a variable appears in the equation determining another, there is vertex between the two nodes.
• Its is then possible to plot the graph and observe the structure of the model.
• In the following graphs, nodes that are part of a system of dependent equations are highlighted in pink.

Example with models SIM-SIMEX

Graph representation of SIM and SIMEX, source: author’s computation

Shock propagation in more complex models

Graph representation of Burgess et al. (2016)

Shock to CAR - Direct

Graph representation of Burgess et al. (2016)

Shock to CAR - Lag 1: Interest rates

Graph representation of Burgess et al. (2016)

Shock to CAR - Lag 2: Income and profits

Graph representation of Burgess et al. (2016)

Dynamical interaction with models

antoinegodin.shinyapps.io/SIMEX

Shiny application

Model with Portfolio choice

Balance Sheet

Transaction Flow Matrix

Behavioural Equations (from PC.txt file)

#######Determination of output - eq. 4.1
y = cons + g
#######Disposable income - eq. 4.2
yd = y - t + r(-1)*b_h(-1)
#######Tax payments - eq. 4.3
t = theta*(y + r(-1)*b_h(-1))
#######Wealth accumulation - eq. 4.4
v = v(-1) + (yd - cons)
#######Consumption function - eq. 4.5
cons = alpha1*yd + alpha2*v(-1)
#######Cash money - eq. 4.6
h_h = v - b_h  
#######Demand for government bills - eq. 4.7
b_h = v*(lambda0 + lambda1*r - lambda2*(yd/v))
#######Supply of government bills - eq. 4.8
b_s = b_s(-1) + (g + r(-1)*b_s(-1)) - (t + r(-1)*b_cb(-1))
#######Supply of cash - eq. 4.9
h_s = h_s(-1) + b_cb - b_cb(-1)
#######Government bills held by the central bank - eq. 4.10
b_cb = b_s - b_h
#######Interest rate as policy instrument - eq. 4.11
r = r_bar

Portfolio Equations

How to allocate wealth between bills and money?

1. Quantity theory of money: links money balances to the flow of income
2. Liquidity preferences: money balances some proportion of total wealth

Merging the two into Tobin’s portfolio equations:

Setting up the environment

Before doing any modelling, we need to load the package in the R environment.

library(PKSFC)

Then, you need to download the attached ‘PC.txt’ file and save it in the folder of your choice. Make sure to set the working directory where you saved the downloaded file. In command line this looks like this but if you use Rstudio, you can use the graphical interface as well (Session>Set Working Directory>Choose Directory)

setwd("pathToYourDirectory")

Loading the model

The first thing to do is to load the model and check for completeness.

pc<-sfc.model("Models/PC.txt",modelName="Portfolio Choice Model")
pc<-sfc.check(pc,fill=FALSE)

Let’s have a look at the graphical representation of model PC. In order to do so, we need to load the Rgraphviz library:

library("Rgraphviz")

We can now look at the graph of model PC:

plot.dag(pc,main="PC" )

You can see that there is a cycle in the graph, implying that GDP, taxes, disposable income and consumption are determined all together (and that they fully adapt to any shock applied to the economy). While this is a mathematical property of the system of equations representing the economy we wish to model, it has an economic meaning and you want to be sure that this is what you believe to be the best representation of what you have in mind.

We are now ready to simulate the model

datapc<-simulate(pc)

Expectations and random shocks

The first of experiment is meant to observe the buffer role of money, in case of random shocks applied to expected disposable income. To do these, we need to modify slightly model pc and change the equations determining consumption, demand for bonds and money, and the expectations on income and wealth. We will call the new model pcRand.

pcRand<-sfc.editEqus(pc,list(list(var="cons",eq="alpha1*yde+alpha2*v(-1)"),
      list(var="b_h",eq="ve*(lambda0 + lambda1*r - lambda2*(yde/ve))")))
pcRand<-sfc.addEqus(pcRand,list(
  list(var="yde",eq="yd(-1)*(1+rnorm(1,sd=0.1))",
         desc="Expected disposable income depending on random shocks"),
  list(var="h_d",eq="ve-b_h",desc="Money demand"),
  list(var="ve",eq="v(-1)+yde-cons",desc="Expected disposable income")))
pcRand<-sfc.editVar(pcRand,var="yd",init=86.48648)
pcRand<-sfc.check(pcRand)

You probably have noticed that the yde equation contains the R function rnorm which allows you to extract a random number from a normal distribution centred around 0, with standard variation of 0.1. The package indeed allows you to use any R function such as min, max or rnorm. However, because of the way the Gauss-Seidel algorithm works, we are facing a problem. Indeed, as the rnorm function extract a number in each iteration, this will mean that the algorithm cannot converge since the difference between each iteration of the algorithm depends of the difference between each extraction. This is why we need to restrict the number of iteration of the Gauss-Seidel.

Before simulating the model, let’s have a look at how the graph of the model has changed:

plot.dag(pcRand,main="PC Random" )

You can now see that the cycle observed in the original model has disappeared. Indeed, consumption doesn’t depend on disposable income any more but on expected disposable income. Let’s now simulate the model.

datapcRand<-simulate(pcRand,maxIter=2)

This replicates figure 4.1, page 110

plot(pcRand$time,datapcRand$baseline[,"h_h"],type="l",xlab="",ylab="",lty=1,lwd=2,
     ylim=c(1*min(datapcRand$baseline[,c("h_h","h_d")],na.rm=T),
            1.2*max(datapcRand$baseline[,c("h_h","h_d")],na.rm=T)))
lines(pcRand$time,datapcRand$baseline[,"h_d"],lty=2,lwd=2)
legend(x=1950,y=1.2*max(datapcRand$baseline[,c("h_h","h_d")],na.rm=T),
       legend=c("Money held","Money demand"),lty=c(2,1),lwd=2,bty="n")

This graph highlights the buffer role of certain stocks in PK-SFC models. Indeed, because expectations are incorrect or because demand might not be equal to supply in any market, at least one stock will not be equal to the targeted level. As highlighted by Foley (1975), in a model without perfect foresight you need a buffer stock in order to obtain equilibrium between demand and supply. The role of buffer stocks in PK-SFC model is thus fundamental and is at the hart of the approach used by Godley (1999) in his seven unsustainable processes. It is by observing the dynamics of certain stock-flow norms that you are able to observe the unsustainable processes evolving in an economy, because stocks precisely absorb disequilibrium.

Interest rates impacts

The graphs presented on the paper are based on the model PCEX, which includes expectation on disposable income and wealth. We will first create the model before simulating it.The shock represents an 100 basis point increase in interest rates.

pcex<-sfc.editEqus(pc,list(
  list(var="cons",eq="alpha1*yde+alpha2*v(-1)"),
  list(var="b_h",eq="ve*(lambda0 + lambda1*r - lambda2*(yde/ve))")))
pcex<-sfc.addEqus(pcex,list(
  list(var="yde",eq="yd(-1)",desc="Expected disposable income"),
  list(var="h_d",eq="ve-b_h",desc="Money demand"),
  list(var="ve",eq="v(-1)+yde-cons",desc="Expected disposable income")))
pcex<-sfc.editVar(pcex,var="yd",init=86.48648)
pcex<-sfc.check(pcex)
init = datapc$baseline[56,]
pcex<-sfc.addScenario(model=pcex,vars="r_bar",values=0.035,inits=1960,
                                            ends=2010,starts=init)
datapcex<-simulate(pcex)

This replicates plot figure 4.3. p. 112

time2=c(1950:2000)
plot(time2, datapcex$scenario[as.character(time2),"h_h"]/
            datapcex$scenario[as.character(time2),"v"],
         type="l",xlab="",ylab="",lty=1,lwd=2)
par(new=T)
plot(time2,datapcex$scenario[as.character(time2),"b_h"]/
            datapcex$scenario[as.character(time2),"v"],
     lty=2, lwd=2,type="l",axes=F,ylab="",xlab="")
axis(4,pretty(c(0.750, 1.1*max(datapcex$scenario[,"b_h"]/
            datapcex$scenario[,"v"],na.rm=T))))
legend(x=1970,y=0.78,legend=c("Share of Bills",
        "Share of Money"),lty=c(2,1),lwd=2,bty="n")
grid()

No surprise here, the change in return rates coming from the shock led to a re-allocation between Bills and Money, in favour of the former.

This replicates fig 4.4 . p 113

time2=c(1950:2000)
matplot(time2,datapcex$scenario[as.character(time2),c("yd","cons")]
                ,type="l",xlab="",ylab="",lty=1:2,col=1,lwd=2)
legend(x=1970,y=88,legend=c("Disposable Income","Consumption"),
             lwd=c(2,2),lty=c(1,2),bty="n")
grid()

These results are more surprising as increased interest rates lead to an increase both in disposable income and consumption. To understand this, one has to bear in mind that increase in interest rates leads to an increase in government spending and thus to an increase in the fiscal stance which determines the steady state:

\(Y^\star=\frac{G+r \cdot B^{\star}_h \cdot (1-\theta)}{\theta}\)

Endogenous propensities to consume

We change the propensities to consume to incorporate the fact that they might be impacted by interest rates and thus have

This piece of code shows how to implement this

pc_end<-sfc.addEqus(pcex,list(
  list(var="alpha1",eq="alpha10-iota*r(-1)")))
pc_end<-sfc.addVars(pc_end,list(
  list(var="alpha10",init="0.7",
         desc="Endoegnous propensity to consume - autonomous term"),
  list(var="iota",init="10",
         desc="Endoegnous propensity to consume - interest rate impact")))
pc_end$scenarios<-NULL
pc_end<-sfc.check(pc_end)
datapc_end<-simulate(pc_end)
init = datapc_end$baseline[56,]
pc_end<-sfc.addScenario(model=pc_end,vars="r_bar",values=0.035,
                                                inits=1960,ends=2010,starts=init)
datapc_end<-simulate(pc_end)

This replicates plot figure 4.9. p. 123

time2=c(1950:2000)
matplot(time2,
     datapc_end$scenario_1[as.character(time2),c("v","yd","cons","y")],
     type="l",xlab="",ylab="",lty=1:4,lwd=2,col=1)
grid()
legend(x=1950,y=100,legend=c("Households wealth","Disposable income",
            "Consumption","GDP"),lty=1:4,lwd=2,bty="n")

This shows that now the short-term dynamics display the Keynesian paradox of thrift but the long-run steady state still shows a positive impact of interests rates on GDP. The short-run recession is governed by the decrease in the propensity to consumed and is slowly compensated by the increase in wealth (and the increased public debt as a counterpart) leading to the new steady state. The steady state is still driven by the fiscal stance.

This replicates plot figure 4.10. p. 123

timelag=c(1950:2000)
time2=c(1951:2001)
plot(time2,
     datapc_end$scenario_1[as.character(time2),c("g")]+
            datapc_end$scenario_1[as.character(timelag),c("r")]*
            datapc_end$scenario_1[as.character(timelag),c("b_s")]-
            datapc_end$scenario_1[as.character(timelag),c("r")]*
            datapc_end$scenario_1[as.character(timelag),c("b_cb")],
     type="l",xlab="",ylab="",lty=1:4,lwd=2,col=1,ylim=c(20.8,23))
lines(time2,datapc_end$scenario_1[as.character(time2),"t"],type="l",xlab="",
            ylab="",lty=2,lwd=2)
grid()
legend("topleft",legend=c("Government expenditures plus net debt services",
            "Tax revenues"),lty=1:4,lwd=2,bty="n")

Debt to GDP at the steady state

The Maastricht treaty: The reference values referred to […] are:

• 3% for the ratio of the planned or actual government deficit to gross domestic product at market
• 60% for the ratio of government debt to gross domestic product at market prices.

Liquidity Preferences and model LP

Value of perpetuity, interest rates, exepcted returns and capital gains

The value of a financial asset is supposed to reflect the net present value of future cash flows. Thus for a long-term bond who is never redeemed, the price \(p_{bL}\) of the bond is given by

\(p_{bL} = \sum \frac{1}{(1+r_{bL})^t} = \frac{1}{r_{bL}}\)

Return rate of an asset is equal to both the yield (interests, dividends) and the capital gains normalized to the nominal value of the asset. In the case of a long-term bond:

\(Rr_{bL} = r_{bL}(-1)+\frac{\Delta p_{bL}}{p_{bL}(-1)} = \frac{1+p_{bL}-p_{bL}(-1)}{p_{bL}(-1)}\)

However, what matter for the households when making their decision is the expected price of bonds \(p_{bL}^e\), given the current price of bonds. This leads to the pure expected rate of return.

\(PERr_{bL} = r_{bL} + \frac{p^e_{bL}-p_{bL}}{p_{bL}}\)

Because expectation are not followed by everyone, households might incorporate a weight \(\xi\) into their expectation formation. This leads to the expected return rate

\(ERr_{bL} = r_{bL} + \xi \frac{p^e_{bL}-p_{bL}}{p_{bL}}\)

The change in prices of bonds leads to capital gains which is one of two parts of the change in nominal value of financial assets. More precisely:

Thus total changes in nominal values of an asset is equal to the quantity of assets bought (or sold) at the current price (\(p_{BL} \cdot \Delta BL\)) and to the capital gains (or losses) made on the stock of assets held at the beginning of the period (\(BL(-1)\cdot \Delta p_{BL}\))

Ostergaard diagram

Balance Sheet

Transaction Flow Matrix

Equation list

# MODEL
# Determination of output - eq. 5.1
y = cons + g
# Regular disposable income - eq. 5.2
yd_r = y - t + r_b(-1)*b_h(-1) + bl_h(-1)
# Tax payments - eq. 5.3
t = theta*(y + r_b(-1)*b_h(-1) + bl_h(-1))
# Wealth accumulation - eq. 5.4
v = v(-1) + (yd_r - cons) + cg
# Capital gains on bonds - eq. 5.5
cg = (p_bl-p_bl(-1))*bl_h(-1)
# Consumption function - eq. 5.6
cons = alpha1*yd_r_e + alpha2*v(-1)
# Expected wealth - eq. 5.7
v_e = v(-1) + (yd_r_e - cons) + cg
# Cash money - eq. 5.8
h_h = v - b_h - p_bl*bl_h
# Demand for cash - eq. 5.9
h_d = v_e - b_d - p_bl*bl_d
# Demand for government bills - eq. 5.10
b_d = v_e*(lambda20 + lambda22*r_b - lambda23*er_rbl - lambda24*(yd_r_e/v_e))
# Demand for government bonds - eq. 5.11
bl_d = v_e*(lambda30 - lambda32*r_b + lambda33*er_rbl - lambda34*(yd_r_e/v_e))/p_bl
# Bills held by households - eq. 5.12
b_h = b_d
# Bonds held by households - eq. 5.13
bl_h = bl_d
# Supply of government bills - eq. 5.14
b_s = b_s(-1) + (g + r_b(-1)*b_s(-1) + bl_s(-1)) - (t + r_b(-1)*b_cb(-1)) -p_bl*(bl_s-bl_s(-1))
# Supply of cash - eq. 5.15
h_s = h_s(-1) + b_cb - b_cb(-1)
# Government bills held by the central bank - eq. 5.16
b_cb = b_s - b_h
# Supply of government bonds - eq. 5.17
bl_s = bl_h
# Expected rate of return on bonds - eq. 5.18
er_rbl = r_bl+chi*(p_bl_e - p_bl)/p_bl
# Interest rate on bonds - eq. 5.19
r_bl = 1/p_bl
# Expected price of bonds - eq. 5.20
p_bl_e = p_bl
# Expected capital gains - eq. 5.21
cg_e = chi*(p_bl_e - p_bl)*bl_h
# Expected regular disposable income - eq. 5.22
yd_r_e = yd_r(-1)
# Interest rate on bills - eq. 5.23
r_b = r_bar
# Price of bonds - eq. 5.24
p_bl = p_bl_bar

Let’s have a look at the graph of model LP

lp<-sfc.model(fileName="Models/LP.txt",modelName="Model Liquidity Preference")
plot.dag(lp)

The adding up of the portfolio behavior

The vertical conditions state that the sum of the exogenous components sums to 1, i.e. you cannot allocate more than the wealth and that the effect of the various return rates and disposable income are subtracting the holding of one asset to allow the increase of the holding of another.

The horizontal conditions, added by Godley (1996) state that the impact of an increase in the ‘own rate’ of an asset should be equal to the impact of a fall in all the other rates.

Other authors propose the symmetry conditions that state that an increase in the return rate of one asset A will have the same impact on the holding of asset B as the increase in the return rate of asset B will have on the holding of asset A

Note that vertical + symmetry imply automatically horizontal but that vertical + horizontal does not necessarily imply symmetry.

Higher interest rates

The first scenario run looks at the impact of interest rates (short- and long-term) on real demand. We assume that the government increases short-term rates from 3% to 4% and the long-term rate from 5% to 6.66%. We assume that this increase is completely unexpected and that the Treasury is believed in that no other change will occur.

lp<-sfc.addScenario(model=lp,vars=list(c("r_bar","p_bl_bar")),values=list(c(0.04,15)),
                            inits=1960,ends=2000)
datalp<-simulate(lp)

This replicates plot figure 5.2. p. 152

time2=c(1950:2000)
plot(time2, datalp$scenario_1[as.character(time2),"v"]/
            datalp$scenario[as.character(time2),"yd_r"],
         type="l",xlab="",ylab="",lty=1,lwd=2)
legend("bottomright",legend=c("Wealth to income ratio"),lty=c(1),lwd=2,bty="n")
grid()

We see that the increase in long-term interest rate leads to capital losses on bonds, leading to a decrease in wealth to income ratio. However because interest rates have increase, households have now a higher disposable income leading to a replenishing of their wealth. Note that as in model PC, the steady state GDP is given by

\(Y^\star=\frac{G+\left(r_B\cdot B^{\star}_h+BL_S^\star\right) \cdot (1-\theta)}{\theta} = \frac{G_{NT}}{\theta}\)

This replicates plot figure 5.3. p. 152

time2=c(1950:2000)
matplot(time2, datalp$scenario_1[as.character(time2),c("yd_r","cons")],
         type="l",xlab="",ylab="",lty=c(1,2), col=1,lwd=2)
legend("topleft",legend=c("Disposable income","Consumption"),lty=c(1,2),lwd=2,bty="n")
grid()

This replicates plot figure 5.4. p. 153

time2=c(1950:2000)
matplot(time2, cbind(datalp$scenario_1[as.character(time2),c("b_h")],
            datalp$scenario_1[as.character(time2),c("p_bl")]*
            datalp$scenario_1[as.character(time2),c("bl_h")])/
                datalp$scenario_1[as.character(time2),c("v")],
         type="l",xlab="",ylab="",lty=c(1,2), col=1,lwd=2)
legend("center",legend=c("Bonds to wealth ratio","Bills to wealth ratio"),
        lty=c(2,1),lwd=2,bty="n")
grid()

Introducing household liquidity

In order to introduce liquidity preferences, we need to modify the model. The first modification is to change the expectation on bonds price to include (i) a learning element and (ii) a stochastic term. Because expectation are going to play a role, we also need to modify the price equation for bonds such that it now reflects a corridor approach to bond pricing. The assumption is thus that the Treasury will try to pin down the price of bonds but only to a certain extent. If the demand is too large (or too low) the treasury will let prices (and interest) float. We assume that the government aims at a certain debt structure in terms of maturity.

#Adding the equations
lp2<-sfc.addEqus(lp,list(
    list(var="z1",equ = "TP>top"),
    list(var="z2",equ = "TP<bot"),
    list(var="TP",equ = "(bl_h(-1)*p_bl(-1))/(bl_h(-1)*p_bl(-1)+b_h(-1))")))
#Adding the parameters
lp2<-sfc.addVars(lp2,list(
    list(var="betae",init=0.5),
    list(var="beta",init=0.01),
    list(var="add",init=0),
    list(var="top",init=0.495),
    list(var="bot",init=0.505)
))
#Adding initial values to the endogenous variables that have lags
lp2<-sfc.editEnd(lp2,var="p_bl_e", init=20)
lp2<-sfc.editEnd(lp2,var="p_bl", init=20)
#Modifying the existing equations
lp2<-sfc.editEqus(lp2,list(
    list(var="p_bl", eq="(1+z1*beta-z2*beta)*p_bl(-1)"),
    list(var="p_bl_e",equ = "p_bl_e(-1)-betae*(p_bl_e(-1) - p_bl) + add")))
#Removing all existing scenarios (coming from LP)
lp2$scenarios<-NULL
#Checking that the model is complete
lp2<-sfc.check(lp2,fill=F)
#Adding the scenario
lp2<-sfc.addScenario(model=lp2,vars=list(c("r_bar")),values=list(c(0.035)),inits=1960,
                                            ends=2000)
lp2<-sfc.addScenario(model=lp2,vars=list(c("add")),values=list(c(-3)),inits=c(1955),
                                            ends=1955)

Let’s have a look at the graph of model LP2

plot.dag(lp2)

We are now ready to simulate the model

datalp2<-simulate(lp2)

This replicates plot figure 5.5. p. 156

time2=c(1950:2000)
plot(time2,datalp2$scenario_1[as.character(time2),"r_b"],
     lty=2, lwd=2,type="l",ylab="",xlab="",ylim=c(0.029,0.036))
par(new=T)
plot(time2, datalp2$scenario_1[as.character(time2),"r_bl"],
         type="l",xlab="",axes=F,ylab="",lty=1,lwd=2,ylim=c(0.0495,0.052))
axis(4,pretty(c(0.0495,0.052)))
legend("bottomright",legend=c("Long term interest rate",
        "Short term interest rate"),lty=c(1,2),lwd=2,bty="n")
grid()

This replicates plot figure 5.6. p. 156

time2=c(1950:2000)
matplot(time2, datalp2$scenario_1[as.character(time2),c("TP","bot","top")],
         type="l",xlab="",ylab="",lty=c(2,1,1), col=1,lwd=2)
legend(x=1950,y=0.502,legend=c("Share of bonds in government debt held by households"),
             lty=c(2),lwd=2,bty="n")
grid()

Let’s add a second scenario by which there is an expected fall in the price of long-term bonds.

lp2<-sfc.addScenario(model=lp2,vars=list(c("add")),values=list(c(-3)),inits=c(1955),
                                            ends=1955)
datalp2<-simulate(lp2)

This replicates plot figure 5.7. p. 157

time2=c(1950:2000)
matplot(time2, datalp2$scenario_2[as.character(time2),c("r_bl")],
         type="l",xlab="",ylab="",lty=c(2,1,1), col=1,lwd=2)
legend(x=1950,y=0.502,legend=c("Share of bonds in government debt held by households"),
             lty=c(2),lwd=2,bty="n")
grid()

This replicates plot figure 5.8. p. 158

time2=c(1950:2000)
matplot(time2, datalp2$scenario_2[as.character(time2),c("p_bl_e","p_bl")],
         type="l",xlab="",ylab="",lty=c(1,2), col=1,lwd=2)
legend("center",legend=c("Expected price of bonds","Actual price of bonds"),
             lty=c(1,2),lwd=2,bty="n")
grid()

This replicates plot figure 5.9. p. 158

time2=c(1950:2000)
matplot(time2, datalp2$scenario_2[as.character(time2),c("TP","bot","top")],
         type="l",xlab="",ylab="",lty=c(2,1,1), col=1,lwd=2)
legend("center",legend=c("Share of bonds in government debt held by households"),
             lty=c(2),lwd=2,bty="n")
grid()

Modelling the firm: price, investment and profits

The role of banks: credit creation and investment

• Firms will always have an investment function determining the desired level of production capacity.
  • Model BMW: \(K^T=\kappa \cdot Y_{-1}\)
• However, the realisation of investment might be constrained by banks credit, leading to an eventual different between desired investment and actual investment.
  • Model BMW: \(I_d = \gamma \cdot (K^T-K_{-1})+DA\) where \(\gamma\) can be interpreted as the outcome between firms and banks interactions.

Functional income distribution

• The usual way to distribute income is between wage and profits (i.e. looking at wage share and profit share).
  • It is however often forgotten that profits are then decomposed further between different type of profits: retained earnings and distributed dividends.
  • But that’s not the end of the story: we can disaggregate even further profits between gross profits and net profits, the difference being 1. interest payments, 2. taxes and 3. capital amortisation
  • It is usually the net profits that are then distributed between retained earnings and dividends.
• In model BMW: depreciation allowances (amortisation) are accounted for explicitly (\(DA= \delta.K_{-1}\)), so are interest payments. We thus have the following functional distribution of income: \(Y=WB+r_{l,-1}\cdot L_{-1} + DA\)
  • Note that in this case, the wage bill is endogenous since both interests payments and depreciation depend only on past values

Consumption function

• In model BWM, the consumption function contains an autonomous term: \(C = \alpha_0+\alpha_1\cdot YD + \alpha_2 \cdot M_{-1}\).
  • The autonomous term plays an important role in the dynamics of the model as it “kick-starts” the model. It plays the same role as government expenditures in the previous model we’ve seen.

Steady state of BMW

• Let’s do it manually, starting from the list of equations:

#Supply of consumption goods - eq. 7.1
 c_s = c_d
#Supply of investment goods - eq. 7.2
 i_s = i_d
#Supply of labour - eq. 7.3
 n_s = n_d
#Transactions of the firms
#GDP - eq. 7.5
 y = c_s + i_s
#Wage bill - eq. 7.6
 wb_d = y - r_l(-1)*l_d(-1) - af
#Depreciation allowances - eq. 7.7
 af = delta*k(-1)
#Demand for bank loans - eq. 7.8
 l_d = l_d(-1) + i_d - af
#Transactions of households
#Disposable income - eq. 7.9
 yd = wb_s + r_m(-1)*m_h(-1)
#Bank deposits held by households - eq. 7.10
 m_h = m_h(-1) + yd - c_d
#The wage bill
#"Supply" of wages - eq. 7.13
 wb_s = w*n_s
#Labour demand - eq. 7.14
 n_d = y/pr
#Wage rate - eq. 7.15
 w = wb_d/n_d
#Household behaviour
#Demand for consumption goods - eq. 7.16
 c_d = alpha0 + alpha1*yd + alpha2*m_h(-1)
#The investment behaviour
#Accumulation of capital - eq. 7.17
 k = k(-1) + i_d - da
#Depreciation allowances - eq. 7.18
 da = delta*k(-1)
#Capital stock target - eq. 7.19
 k_t = kappa*y(-1)
# below is an additional equation I've defined to get output to capital ratio as in figure 7.4, the model will run even without equation
OCR = y/k(-1)  
#Demand for investment goods - eq. 7.20
 i_d = gamma*(k_t - k(-1)) + da

• Important points to discuss:
  • Role of autonomous consumption
  • The existence of non-negative GDP steady state
  • The equality between capital stock and wealth

Out of equilibrium values

• To do so, let’s have a look at the graph representation of BMW:

BMW<-sfc.model("Models/BMW.txt")

## Warning in sfc.check(model, fill = fill): The following variables have lags
## but no initial values: - r_l - r_m

plot.dag(BMW)

• We need to solve the pink cycle and express Y (and K) as a function of only lagged variables.

Stability

• Very brief intro because stability analysis of difference system is much more complicated than for differential systems.
• The stability of a system of equations \(z_t = A \cdot z_{t-1} + c\) can be analysed via the trace, determinant and discriminant of A
  • trace = sum of diagonal terms, for 2x2 matrix: \(tr_A = a_{11}+a_{22}\)
  • determinant, for a 2x2 matrix: \(det_A = |A| = a_{11}\cdot a_{22} - a_{21}\cdot a_{12}\)
  • discriminant: \(\Delta = tr_A^2+4\cdot det_A\)
• Necessary condition for stability is that they absolute value of the determinant is smaller than 1
• If this is respected, then either the determinant is positive or if it is negative, the trace has to be smaller than 2 if and only if \(-det_a<1-|tr_A|\).
• We won’t go in the details, but the analysis highlight some important characteristics:
  • the system will converge provided the marginal propensity to save is higher than the marginal propensity to invest

More on profits and inventories

• Inventories are fundamental for the working of a firm in terms of production smoothing, response to sudden changes in demand and dealing with uncertainty. They are very much related to the notion of time in SFC (and other types of post-Keynesian) models. Time is historical, based on past observation and projected in the future by expectations, on the contrary of mainstream models where the future is pulled in the present.
• Example of a current account matrix for firms

• Entrepreneurial vs. total profits
  • Entrepreneurial profits: \(F=S-(WB-\Delta IN + r_{l,-1} IN_{-1})\)
  • Total profits: \(F=S-(WB-\Delta IN)\)
• Historical wage costs vs. total historic costs
  • Historical wage costs: \(HWC = (WB-\Delta IN)\)
  • Total historical costs: \(HC = (WB-\Delta IN)+r_{l,-1} IN_{-1}\)
• We can show that profits can always be expressed as a fraction of historical costs: \(F= \phi' HC\), implying that sales can also be expressed as a function of historical costs only
  • \(S = F + HC\)
  • \(S = (1+\phi')HC\)
  • But Sales are also \(S = s.p\), we can thus show that \(p = (1+\phi')HUC\) where \(HUC=\frac{HC}{s}\) is the historical unit cost.
• Important distinction between entrepreneurial profits and cash flows and the definition of national accounts profits
  • Entrepreneurial profits: \(F=S-(WB-\Delta IN + r_{l,-1} IN_{-1})\)
  • Cash flow: \(CF=S - WB - r_{l,-1} IN_{-1}\)
  • NIPA profits: \(F_{nipa}=S - WB + \Delta IN - \Delta UC in_{-1}\) in order to remove stock appreciation or inventory valuation adjustment which are not “proper flows”